Erlang-delayed stochastic chemical kinetic formalism for efficient analysis of biological systems with non-elementary reaction effects

Stochastic chemical kinetics (SCK) has become an impor- tant formalism for modeling and analysis of complex biological systems as it can capture the discreteness and the randomness of underlying biochemical reactions. One of the assumptions made by SCK is that each reaction be an elementary step which cannot be broken down into smaller steps. As such, transition events of the SCK model occur spontaneously in that there is no time lag between the reaction initiation and completion. In practice, however, it is very difficult to experimentally determine if an observed state change is a result of an elementary reaction or a sequence of several reaction steps. To test various hypotheses on such time-delays through the use of the SCK, all of the intermediate reactions and species need to be explicitly specified, which can quickly become cumbersome. To more efficiently model and analyze potential effects of such intermediate reaction steps, this paper proposes a new for-malism for higher-level discrete-stochastic treatment of biological systems with reaction delays. Our new formalism can represent a time delay caused by intermediate reaction steps as an Erlang random variable, allowing a model's size to be reduced substantially. This paper illustrates an application of this formalism for analysis of the effect of different transcription elongation steps and rates on the distribution of RNA molecules.